Table 3: Approximation of the energy levels of the harmonic oscillator in dimensions . represents the energy state, is the exact solution for the energy given by (12), and is the upper bound to obtained using the present variational analysis. The eigenvalues of H are minimized over .


3
1 3 3.00000000 6.00
0 5 19 19.00000001 7.00
10 39 39.00000001 8.75

1 5 5.00007348 4.50
1 5 21 21.00167944 6.25
10 41 41.00907276 7.75

1 7 7.00000000 6.00
2 5 23 23.00000001 7.75
10 43 43.00000001 9.25

1 9 9.00000001 6.00
3 5 25 25.00000076 7.25
10 45 45.00002070 8.50

4
0
1 4 4.00073469 4.25
5 20 20.00745550 6.00
10 40 40.02454449 7.50
1
1 6 6.00000262 5.00
5 22 22.00011370 6.50
10 42 42.00094014 8.00
2
1 8 8.00000002 6.00
5 24 24.00000248 7.25
10 44 44.00004592 8.50
3
1 10 10.00000000 6.00
5 26 26.00000008 7.50
10 46 46.00000274 8.75

5
0
1 5 5.00007348 4.50
5 21 21.00167944 6.25
10 41 41.00907276 7.75
1
1 7 7.00000000 6.00
5 23 23.00000001 7.75
10 43 43.00000001 9.25
2
1 9 9.00000001 6.00
5 25 25.00000076 7.25
10 45 45.00002070 8.50
3
1 11 11.00000001 6.00
5 27 27.00000000 8.00
10 47 47.00000001 10.00